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what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.

So far, I have not found any precise definition in literature.

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Have you looked in Wikipedia (or in a textbook about PDEs, for that matter)? In what way are you not satisfied by the definitions given there? –  Mariano Suárez-Alvarez Feb 11 '11 at 14:33
    
@Mariano: most textbooks do not give satisfactory answer above second order... –  Willie Wong Feb 11 '11 at 15:15
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@doraemonpaul Please stop unnecesarily retaggin old questions, you're clumping the main page. –  Pedro Tamaroff Aug 21 '12 at 1:45
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1 Answer 1

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The short answer is: there is none.

The slightly less short answer is: there are some nice descriptions of ellipticity for higher order equations and systems. There are some nice descriptions of hyperbolicity for higher order equations and systems. There is generally no description for parabolicity. And elliptic, parabolic, and hyperbolic do not come close to exhausting the possible PDEs that can be written down beyond second order scalar equations.

The long answer is, well, long.


To scratch at the surface of this problem, we need to journey back to 1926, when the definitions of elliptic, parabolic, and hyperbolic PDEs are given by Jacques Hadamard. In his study of scalar linear partial differential equations of second order (the work has since been compiled and published as Lectures on Cauchy's problem in linear partial differential equations by Dover publications in 1953), Hadamard made the following definitions. (As an aside, it is also in those lectures that Hadamard made the first modern definition of well-posedness of the Cauchy problem of a PDE.)

Given a linear partial differential operator of second order with real coefficients, its principal part can be represented by a (symmetric) matrix of coefficients $a^{ij}\partial^2_{ij}$. The operator is then said to be

  • Elliptic if $a^{ij}$ is positive definite or negative definite
  • Parabolic if $a^{ij}$ is positive or negative semi-definite, and admits precisely one 0 eigenvalue.
  • Hyperbolic if $a^{ij}$ is indefinite, but is non-degenerate; that is, for every vector $v_j$, there exists some vector $w_i$ such that $w_ia^{ij}v_j \neq 0$. In other words, it has no zero eigenvalue.

Already you see that the classification is incomplete: any operator with principle part having nullity $>1$, or having nullity $= 1$ but with indefinite sign is not classified. Furthermore, the definition of hyperbolic is different from the modern one. Indeed, Hadamard made the additional definition that the operator is

  • Normal hyperbolic if $a^{ij}$ is hyperbolic, and furthermore all but one of its eigenvalues have the same sign.

The wave operator is a normal hyperbolic operator. Whereas what now-a-days we call the ultrahyperbolic operators (in order to distinguish them from hyperbolic ones) were nominally considered to be hyperbolic by Hadamard's standards.

Hadamard was able to show that linear, normal, hyperbolic PDEs admit well-posed initial value (Cauchy) problems for data prescribed on hypersurfaces over which tangent space the restriction of $a^{ij}$ is elliptic.

Hyperbolicity redefined

Let us first talk about hyperbolic equations (because I'm more familiar with its history).

The next advance came around the mid 20th century, with contributions by people like Jean Leray and Lars Gårding. Rather than using Hadamard's geometric definition of hyperbolicity, in order to deal with higher order equations and systems of equations, hyperbolicity becomes more-or-less synonymous with "having a well-posed initial value problem". This recognition does not mean the definition is separate from the geometry of the coefficients. Here I give a theorem (which you can find in Lars Hörmander's Analysis of Linear Partial Differential Operators):

Theorem A linear partial differential operator of order $m$ with smooth coefficients admit a well-posed Cauchy problem for arbitrary smooth data if and only if its symbol is a hyperbolic polynomial.

Basically, a linear partial differential operator $P = \sum_{|\alpha| \leq m} (i)^{|\alpha|}a^{\alpha}\partial_\alpha$ (where $\alpha$ is a multi index) has corresponding symbol $p(\xi) = \sum_{|\alpha| \leq m} a^\alpha \xi_\alpha$. The symbol $p$ is said to by a hyperbolic polynomial in $\xi\in\mathbb{R}^n$ if there exists a vector $\tau\in\mathbb{R}^n$ such that the $\sum_{|\alpha| = m} a^\alpha \tau_\alpha \neq 0$, and that there exists some number $s_0 > 0$ such that for all $s < s_0$, the polynomial $$ p(\xi + i s \tau) \neq 0 \qquad \forall \xi\in\mathbb{R}^n $$

To apply to a system of equations, We consider the system given by

$$ P_{ij}u^j = 0 $$

and $P_{ij} = \sum (i)^{|\alpha|} a^{\alpha}_{ij} \partial_\alpha$ is a matrix of scalar partial differential operators, the condition is analogous, with the symbol defined instead by

$$ p(\xi) = \det( \sum_{|\alpha| \leq m} a^{\alpha}_{ij} \xi_\alpha ) $$

via the determinant.

The problem with this definition is that it is not-so-easy to check, and does not generalize easily to non-linear equations.

As it turns out, to consider quasilinear equations, one realizes that what is really important for the well-posedness of the initial value problem is only the principal part of the operator (that is, the part with the highest number derivatives). There have been many attempts to provide sufficient conditions on the principal part of the operator that guarantees the well-posedness of the Cauchy problem, and all of these conditions are named as "(blank) hyperbolicity". Notice that none of them are necessary conditions: they are all strictly stronger assumptions than the conditions given above in the theorem.

I will not attempt to give all the definitions here. I'll refer you to some keywords for searching in textbooks or literature. First there is the notion of strict hyperbolicity. This is the direct analogue of the above characterisation using hyperbolic polynomials in the context of principal symbols. In fact, we have that

Theorem If the principal part of an operator is strictly hyperbolic, than regardless of what the lower order terms are, the operator is hyperbolic.

The definition of strict hyperbolicity is as follows: construct the principal part of the symbol as above, but just using the highest order terms $p_m(\xi) = \sum_{|\alpha| = m} a^\alpha\xi_\alpha$, so it is a homogeneous polynomial. It is strictly hyperbolic if there against exists some vector $\tau$ such that for any $\xi\neq 0$, the $m$-th degree polynomial in $s$ given by $p(\xi + s\tau)$ has $m$ distinct real roots.

A rather different definition of hyperbolicity is called symmetric hyperbolicity. This relies on re-writing your higher-order equation/system as a first order system of conservation laws. It requires being able to represent/convert the system into a special algebraic form, with certain symmetry and positivity conditions on its principal part. There are many, many good books written about this subject.

One interesting facet of symmetric hyperbolic systems is that sometimes the usual/simple way of reducing a system does not suffice to exhibit the symmetric hyperbolic structure, and one may be forced to "augment" the system with dummy dependent variables. Much of this is discussed in C. Dafermos's book Hyperbolic conservation laws in continuum physics, and has been used by, e.g. Denis Serre to show that complicated systems such as the Maxwell-Born-Infeld model of non-linear electrodynamics is hyperbolic. (Though the same can be shown, perhaps slightly more easily, using the next method; a proof can be found in Jared Speck's recent paper.)

A third method of defining hyperbolicity is the notion of regular hyperbolicity due to Demetrios Christodoulou in his book Action Principle and Partial Differential Equations (though the germ of the idea was already considerd by Hughes, Kato, and Marsden in the 1970s). This is a method specially adapted for studying second order quasilinear systems of differential equations. I gave a quick discussion in a recent paper of mine. In particular, it is noted in that paper that there is a large gap between regular hyperbolicity, which is perhaps the least restrictive in the context of second order systems of equations of the three above, and ellipticity.

There are also other more special cases where the "time direction" is distinguished in the system of equations. In particular there is a large class of equations of the form

$$ \partial_t^{2m}u = F(u,\partial u, \ldots, \partial^{2m-1} u) + A u $$

which can easily to be seen as hyperbolic as long as $A$ is a quasilinear elliptic operator of order $2m$.

In general a principal feature of hyperbolic equations is that their solution tends to rely on the existence of $L^2$ energy estimates.

Ellipticity

Compared to hyperbolicity, ellipticity is better understood (but unfortuantely, not by me. So I'll point you to references).

For scalar nonlinear equations, there are very well developed theory available in various textbooks. See, for example, Gilbarg and Trudinger, Elliptic partial differential equations of second order (pay attention to the last few chapters), or perhaps Caffarelli and Cabre, Fully nonlinear elliptic equations for development in the second order case. For higher order operators, say $P$ is of order $2m$, a genearally used criterion is that its principal symbol satisfies the elliptic inequality $p_{2m}(\xi) \geq \lambda |\xi|^{2m}$.

For systems, on the other hand, ellipticity is also a tricky business. In so far as second order systems are concerned, a very useful condition is the so-called Legendre-Hadamard condition, otherwise known as rank-1 convexity condition. This condition goes back to (if I am not mistaken) Jesse Douglas and C.B. Morrey in their works on the Plateau problem in Riemannian geometry. The effectiveness of this condition is still a much studied subject related to nonlinear elasticity and calculus of variations. You can learn quite a bit about it by looking at some of John Ball's survey articles.

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Ah, forgot to include the following in the text: there are, as far as I know, no completely general method of defining fully nonlinear hyperbolic equations. This has some relation to the fact that solving the initial value problem often uses energy estimates. On the other hand, one can easily state what it means for a fully nonlinear equation to be elliptic. See the wikipedia page on elliptic operator, and also search the web for Monge-Ampere equation if you are interested. –  Willie Wong Feb 11 '11 at 16:39
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Very nice answer! (I added an å and corrected a typo) –  Mariano Suárez-Alvarez Feb 11 '11 at 17:02
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I agree! (And speaking of Swedish names, "Hormander" should be Hörmander.) –  Hans Lundmark Feb 11 '11 at 19:00
    
@Mariano @Hans: I'm dangerously close to CW'ing this answer based on number of edits. So before I throw in the "ö" edit, I want to double check to see if there's anything else I want to change. Thanks for keeping me honest! :) –  Willie Wong Feb 11 '11 at 19:15
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@Willie Wong, You're awesome. –  BBischof Jun 9 '11 at 18:54
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